The PPT-squared conjecture: is the composition of two PPT channels always entanglement breaking?

Theorem 1.3: there is an explicit trace-preserving PPT channel $\Theta$ on $M_{21}(\mathbb C)$ with $\Theta\circ\Theta$ not entanglement breaking, answering the 2012 channel question negatively with the same channel in both slots. Theorem 1.2: explicit PPT maps $\Phi_1,\Phi_2$ on $M_{10}(\mathbb C)$ (not trace preserving) whose composition has nonzero Choi matrix with no product vector in its range, refuting the unrestricted two-map Conjecture IV.1. The same pair yields the zero-key entangled state of the companion entry. Not shown: minimal dimensions, or whether every PPT channel becomes entanglement breaking after three compositions (the PPT-cubed question).

Theorem 1.3: there is an explicit trace-preserving PPT channel $\Theta$ on $M_{21}(\mathbb C)$ with $\Theta\circ\Theta$ not entanglement breaking, answering the 2012 channel question negatively with the same channel in both slots. Theorem 1.2: explicit PPT maps $\Phi_1,\Phi_2$ on $M_{10}(\mathbb C)$ (not trace preserving) whose composition has nonzero Choi matrix with no product vector in its range, refuting the unrestricted two-map Conjecture IV.1. The same pair yields the zero-key entangled state of the companion entry. Not shown: minimal dimensions, or whether every PPT channel becomes entanglement breaking after three compositions (the PPT-cubed question).